Finite Group Databasebeta

← #208447

Subgroups — #208447

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35086C211yesmaximal, characteristic
2#21904C211yesmaximal, characteristic
3#13749C211yesmaximal, characteristic
4#9910C212yesmaximal, complemented
5#9711C212yesmaximal, complemented
6#9146C212yesmaximal
7#9143C212yesmaximal
8#8219C212yesmaximal, complemented
9C4×#328C212yesmaximal, complemented
10#2857C2211yescharacteristic
11#2851C2211yescharacteristic
12#2848C2212yes—
13#2652—22no—
14#2505—22no—
15#2496C412yescomplemented
16#2489C412yescomplemented
17#2480—22no—
18#2429C412yescomplemented
19#2421C412yescomplemented
20#2334—22no—
21C4×C22⋊Q8C2212yes—
22C2×C4×C22⋊C4—22no—
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C22×#95C2211yescharacteristic
26#1924—22no—
27#1923—22no—
28#1797C2212yes—
29#1779C2212yes—
30#1749C2211yescharacteristic
31C4×C22⋊C8C2211yescharacteristic
32C2×#328C2212yes—
33C2×#328C2212yes—
34C2×#328C2212yes—
35C2×#328C2212yes—
36C4×#95—22no—
37C4×#95C2212yes—
38C4×#95C2212yes—
39#1294C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic, complemented
46C2×#126—22no—
47C2×#125C2×C411yescharacteristic, complemented
48C2×C22⋊Q8C2×C412yes—
49C2×C22⋊Q8C2×C412yes—
50C2×C22⋊Q8C2312yes—
51C2×C4×Q8C2×C412yescomplemented
52C22×C22⋊C4—21no—
53C22×C22⋊C4D412yes—
54C22×C42—21no—
55C22×C42—22no—
56C22×C42—22no—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C4⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8C2312yes—
63C2×C22⋊C8C2312yes—
64C2×#97—21no—
65C2×C4×C8—21no—
66#400—22no—
67#400—22no—
68#400—22no—
69#398—22no—
70#397—22no—
71#397—22no—
72#397—22no—
73#396—22no—
74#391C2×C412yescomplemented
75#391C2×C412yescomplemented
76#389—22no—
77#389—22no—
78#389C2×C412yescomplemented
79#389C2×C412yescomplemented
80#388—22no—
81#388—22no—
82#388—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#385—22no—
88#385—22no—
89#385—22no—
90#385—22no—
91#385—22no—
92#384C2×C411yescharacteristic, complemented
93#382—22no—
94#382C2×C411yescharacteristic, complemented
95#380—22no—
96#380C2×C412yes—
97#380C2×C412yes—
98#380C2312yes—
99#379—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4C2×C412yescomplemented
102C4×C22⋊C4—216no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4C2×C412yes—
107C4×C22⋊C4D412yes—
108C4×C22⋊C4D412yes—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—24no—
113C2×#95D412yes—
114C2×#95C2312yes—
115C2×#95C2312yes—
116#336—22no—
117#336—21no—
118#336—21no—
119#328—24no—
120#328—24no—
121#328—24no—
122#328—24no—
123C25C2×D411yescharacteristic
124C22×Q8C22×C412yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—28no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4—82no—
134C23×C4—81no—
135C23×C4—81no—
136C23×C4C22×C412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C4⋊Q8—81no—
145#126—48no—
146#125—81no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—82no—
152C22⋊Q8—82no—
153C4×Q8—82no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C42—22no—
180C2×C42—28no—
181C2×C42—28no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—81no—
185C2×C42—81no—
186C2×C42—82no—
187C2×C42—82no—
188C2×C42—82no—
189C2×C42—82no—
190C2×C42—81no—
191C2×C42—81no—
192C2×C42C22×C412yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C4⋊C8—82no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—82no—
204C22⋊C8—82no—
205#97—81no—
206C4×C8—81no—
207#95—24no—
208#95—28no—
209#95—28no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95C22⋊C412yes—
224#95C22⋊C412yes—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227#95#4114yes—
228#95#4114yes—
229C24—21no—
230C24—21no—
231C24—24no—
232C24—22no—
233C24—22no—
234C24—41no—
235C24—41no—
236C24C2×C22⋊C412yes—
237C24C22×D411yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8—42no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—216no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—84no—
282C22×C4—84no—
283C22×C4—88no—
284C22×C4—88no—
285C22×C4—84no—
286C22×C4—84no—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C22⋊C4—216no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4#10418yes—
329C42—41no—
330C42—41no—
331C42—42no—
332C42—42no—
333C42—41no—
334C42—41no—
335C42—44no—
336C42—82no—
337C42—88no—
338C42—88no—
339C42—84no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—24no—
344C23—24no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23C2×#10414yes—
361C23C22×C22⋊C411yescharacteristic
362C23#52511yescharacteristic
363C23#52511yescharacteristic
364C23#57712yes—
365C23#57812yes—
366Q8—82no—
367Q8—82no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—816no—
422C2×C4C2×#10414yes—
423C2×C4C2×#10414yes—
424C2×C4C4×#4112yes—
425C2×C4#53212yes—
426C8—82no—
427C8—82no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—44no—
440C22—48no—
441C22—44no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22#229711yescharacteristic
447C22#288311yescharacteristic
448C22#288311yescharacteristic
449C22#289812yes—
450C22C22×#10412yes—
451C22#312012yes—
452C22#312311yescharacteristic
453C22#312711yescharacteristic
454C4—24no—
455C4—24no—
456C4—22no—
457C4—22no—
458C4—48no—
459C4—48no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—82no—
464C4—82no—
465C4—82no—
466C4—82no—
467C2—21no—
468C2—22no—
469C2—21no—
470C2—44no—
471C2#1754812yes—
472C2#1754812yes—
473C2#2139811yescharacteristic
474C2#2141511yescharacteristic
475C2#2142211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9910C2semidirect
5. #9711C2semidirect
8. #8219C2semidirect
9. C4×#328C2semidirect
15. #2496C4semidirect
16. #2489C4semidirect
18. #2429C4semidirect
19. #2421C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
47. C2×#125C2×C4semidirect
51. C2×C4×Q8C2×C4semidirect
74. #391C2×C4semidirect
75. #391C2×C4semidirect
78. #389C2×C4semidirect
79. #389C2×C4semidirect
92. #384C2×C4semidirect
94. #382C2×C4semidirect
101. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#350862
#219042
#137492
#99102
#97112
#91462
#91432
#82192
C4×#3282